Odd-Number Cyclo[n]Carbons Sustaining Alternating Aromaticity

Cyclo[n]carbons (n = 5, 7, 9, ..., 29) composed from an odd number of carbon atoms are studied computationally at density functional theory (DFT) and ab initio complete active space self-consistent field (CASSCF) levels of theory to get insight into their electronic structure and aromaticity. DFT calculations predict a strongly delocalized carbene structure of the cyclo[n]carbons and an aromatic character for all of them. In contrast, calculations at the CASSCF level yield geometrically bent and electronically localized carbene structures leading to an alternating double aromaticity of the odd-number cyclo[n]carbons. CASSCF calculations yield a singlet electronic ground state for the studied cyclo[n]carbons except for C25, whereas at the DFT level the energy difference between the lowest singlet and triplet states depends on the employed functional. The BHandHLYP functional predicts a triplet ground state of the larger odd-number cyclo[n]carbons starting from n = 13. Current-density calculations at the BHandHLYP level using the CASSCF-optimized molecular structures show that there is a through-space delocalization in the cyclo[n]carbons. The current density avoids the carbene carbon atom, leading to an alternating double aromaticity of the odd-number cyclo[n]carbons satisfying the antiaromatic [4k+1] and aromatic [4k+3] rules. C11, C15, and C19 are aromatic and can be prioritized in future synthesis. We predict a bond-shift phenomenon for the triplet state of the cyclo[n]carbons leading to resonance structures that have different reactivity toward dimerization.


I. INTRODUCTION
Cyclo[n]carbons represent a unique and underexplored form of carbon molecules among numerous other allotropes. 1 They have been observed in gas-phase mass-spectra experiments. 2−6 The first ever purposeful chemical synthesis of an even-number cyclo [18]carbon (C 18 ) was carried out by Kaiser et al. in 2019, 7 while last year, Scriven et al. improved the synthetic protocol for cyclo [18]carbon, reaching a reaction yield of 64%. 8 After these successful synthesis endeavors, a large number of articles have been devoted to studies of the electronic structure of cyclo[n]carbons. Most of them focus on the even-number members of this class of molecules. 9−21 Studies on odd-number cyclo[n]carbons have been limited to very few articles that mainly report properties of small carbon rings up to C 17 calculated at the DFT level of theory. 22−26 These articles were mainly aimed to describe the relative stability of linear vs cyclic isomers. The aromaticity and electronic properties of odd-number cyclo[n]carbons were studied for the first time in 2009 by Fowler et al. at the B3LYP/6-31G(d) level of theory. 27  electrons belong to the in-plane π-system. As a result, they found that all odd-number cyclo[n]carbons sustain a diatropic ring current in the out-of-plane π-system, which is stronger than the paratropic ring current in the in-plane π-system, i.e., they found that the studied odd-number cyclo[n]carbons (C 7 − C 29 ) are globally aromatic and called them "π aromatic and σ antiaromatic". 27 Here, we challenge this conclusion since the employed DFT calculations are not able to properly describe the carbene structure of the odd-number cyclo[n]carbons. Recently, Seenithurai and Chai 28 systematically studied oddnumber cyclo[n]carbons containing up to 99 carbon atoms at the TAO-DFT-LDA level of theory and showed that some electronic descriptors like the ionization potential, electron affinity, and the optical gap alternate with n. They did not obtain any clear oscillations in the singlet−triplet energy gap (ΔE ST ) but reported ΔE ST of 4−5.6 kcal mol −1 for C 13 −C 49 , which is remarkable when considering the polyradical nature of these species. As Seenithurai and Chai 28 unfortunately did not provide any analysis of the optimized molecular structures of the studied odd-number cyclo[n]carbons, we cannot judge how well their structures agree with the ones we report here. Hobza et al. 29 predicted very recently a triplet ground state for C 17 based on ωB97XD/def2-TZVPP calculations, which is most likely incorrect due to the limitations of the employed DFT level to describe the localized carbene structure of odd-number cyclo[n]carbons. Here, we report and analyze the electronic and molecular structures of odd-number cyclo[n]carbons obtained at the CASSCF level of theory. Current densities are calculated at the DFT level using the BHandHLYP functional to understand the electronic structure properties and the aromatic character of cyclo[n]carbons (n = 5, 7, 9, ..., 29). We utilize the gauge-including magnetically induced currents (GIMIC) method 30,31 for analyzing the aromatic properties. Based on these calculations, we propose a novel concept of through-space aromaticity of the odd-number cyclo[n]carbons in the singlet ground state, which is schematically presented in Figure 1. The calculations show that the electron delocalization avoids the empty p out orbital at the carbene atom, leading to alternating aromaticity of the odd-number cyclo[n]carbons. The cyclo[n]carbons with [4k +1] carbon atoms are antiaromatic, whereas those with [4k+3] carbon atoms are aromatic, where k is a positive integer.

II. COMPUTATIONAL DETAILS
The molecular structures of the lowest singlet and triplet states of cyclo[2n+1]carbons (n = 2−14) were initially optimized at the DFT level using the hybrid functional BHandHLYP 32,33 and the Karlsruhe def2-TZVP basis set. 34 Since previous studies of the electronic structure of even-numbered cyclo[n]carbons showed that the experimentally observed polyyne structures are obtained only when using a functional with a large amount of the Hartree−Fock exchange (>40%), 10,15 we used here the BHandHLYP functional that satisfies this condition with 50% of the Hartree−Fock exchange. We also optimized the molecular structures using functionals with different amounts of Hartree−Fock exchange: TPSSh (0%), 35 B3LYP (20%), 33,36 BMK (42%), 37 and M06HF (100%). 38 The molecular structures of the lowest singlet and triplet states were also optimized at the ab initio complete active space self-consistent field (CASSCF) level, 39,40 since multiconfiguration SCF calculations yielded the experimentally observed polyyne structure for the even-numbered cyclo[n]carbons. 10 The 6-31G(d,p) 41,42 basis set was used in the optimization of the molecular structures at the CASSCF level. We used the (14,12) active space (14 electrons in 12 active orbitals) for all of the studied molecules. Smaller and larger active spaces were also used for checking the choice of the active space. The structure optimizations at the CASSCF level were performed using the Firefly software. 43,44 We also performed structure optimizations of the lowest singlet and triplet states of C 11 at the multireference second-order perturbation theory (CASPT2) level using split-valence polarization basis sets (SVP) 45,46 with the BAGEL program. 47 Calculations of the magnetically induced current densities were carried out using the gauge-including magnetically induced currents (GIMIC) method. 48,49 The basis-set information, the atomic orbital density matrix, and the perturbed atomic orbital density matrices are the input data of the GIMIC calculations. The density matrices are obtained by performing NMR shielding calculations.
The NMR shielding calculations were performed at the BHandHLYP/def2-TZVP level using the BHandHLYP and CASSCF structures. The first-order magnetically perturbed density matrices and the DFT-optimized molecular structures of the cyclo[2n+1]carbons (n = 2−14) were computed using the Gaussian 16 program package. 50 A script was used for converting the Gaussian output to GIMIC input data in appropriate format. 48,49 The strengths of the magnetically induced ring currents (I in nA T −1 ) were obtained by integrating the current-density flux that passes through a plane placed perpendicular to the molecular plane. The visualization of the current densities was performed within the Multiwfn software. 51 We also calculated NMR shielding tensors at CASSCF/SVP 45,46 level of theory for singlet and triplet states of the C 11 and C 13 cyclocarbons using DALTON software. 52 The ring-current strengths were then obtained at the CASSCF level by integrating the zz component of the shielding tensor along the symmetry axis in the middle of the ring using the Ampere−Maxwell's law. 53

III. RESULTS AND DISCUSSION
III.I. Electronic Structure of Even-Number Cyclocarbons. The even-number cyclo[n]carbons are doubly aromatic according to the Huckel rules for antiaromatic [4k] and aromatic [4k+2] rings with k = 1−8. 9 The aromaticity is maintained by the delocalized in-plane (p in ) and out-of-plane (p out ) orbitals, respectively, 9 which can also be interpreted as double σ (in-plane) and π (out-of-plane) aromaticity and antiaromaticity of the [4k+2] and [4k] species. We recently showed that doubly aromatic C 18 sustains a stronger magnetically induced ring current (MIRC) in the 18π out outof-plane electrons (I out = 21.8 nA T −1 ) than in the 18π in in-theplane electrons (I in = 7.2 nA T −1 ) due to the perpendicular orientation of the p out atomic orbitals (AO) with respect to the magnetic field direction. The radial p in AOs have an external CCC angle of 200°affecting the electron delocalization. 10 For k > 8, the even-number cyclo[n]carbons have a pronounced bond alternation with a transition to nonaromaticity occurring for the [4k] and [4k+2] species. 9 The larger even-number cyclo[n]carbons do not sustain any MIRC. The results of the GIMIC calculations agree completely with the results obtained with simple independent-particle simulations by Bylaska et al. 54 III.II. Electronic Structure of Odd-Number Cyclocarbons. The odd-number cyclo [n]carbons have an extra carbon atom (a carbene) affecting their electronic structure. It is well known that carbenes generally have a triplet ground state with a nonlinear molecular geometry. 55,56 Thus, a triplet ground state can be anticipated also for odd-number cyclo[n]carbons, and indeed the BHandHLYP/def2-TZVP calculations suggest that C 13 and larger odd-number cyclo[n]-  19 (right). The net antiaromatic C 17 is π antiaromatic and σ aromatic and vice versa for the net aromatic C 19 . Red and blue arrows denote the paratropic and diatropic currents, respectively; π in and π out mean that the electrons are aligned in molecular plane and perpendicular to it, respectively.
carbons have a triplet ground state. For C 13 , the triplet state is 8.5 kcal mol −1 below the singlet state. The energy difference between the singlet and triplet states (ΔE ST ) increases for larger rings and reaches 21.2 kcal mol −1 for C 29 (see Table 1). Calculations on C 17 employing DFT functionals with different amounts of Hartree−Fock exchange (from 0 to 100%) do not qualitatively change the trend; see Table S1.
However, calculations at the CASSCF level with an active space consisting of 14  The MIRC has also been calculated at other levels of theory. b I(S) and I(T) calculated at the BHandHLYP/6-31G(d,p) level using the CASSCFoptimized geometries. c I(S) and I(T) calculated at the CASSCF/SVP level using the CASSCF-optimized geometries. d I(S) and I(T) calculated at the M06-2X/6-31G(d,p) level using the CASSCF-optimized geometries.   58 and ωB97XD 59 functionals qualitatively yield the same trend for ΔE ST as obtained at the CASSCF level (see Table  S2). The calculations thus show that different trends for ΔE ST are obtained when using the DFT-optimized structures and the CASSCF ones.
Despite the fact that BHandHLYP and M06-2X functionals give quantitatively different values for the ΔE ST gap (Table S2) when using CASSCF geometries, the strength of the MIRC for the S and T states calculated with these functional is largely the same ( Table 1), implying that the strength of the MIRC values is almost independent of the employed level of DFT theory, while it strongly depends on the employed molecular structure. There is only a qualitative correlation between the ΔE ST gap and the MIRC strength. An alternating S−T gap leads to a similar alternation in the aromatic/antiaromatic behavior of the odd-number cyclocarbons (Tables 1 and S2).
III.III. Structural Features of Odd-Number Cyclocarbons. The molecular structures optimized at the BHandHLYP and CASSCF levels are shown in Table S3. The optimized structures at the two levels of theory are very different even though both levels predict a polyyne-type structure with alternating CC bonds. The BHandHLYP/6-31G(d,p) calculations yield a strongly delocalized carbene structure for the singlet (S) and triplet (T) states, i.e., the lonepair electrons are not localized to a single carbon atom, resulting in short CC bonds (d 1 and d 2 ) near the carbene center and an obtuse CCC angle (θ) (Figure 2). Similar structural trends are obtained for the θ angle and the d 1 and d 2 bond lengths of the singlet and triplet states with increasing ring size. The strong lone-pair delocalization and the bondlength equalization obtained at the DFT level lead to a triplet ground state for most of the studied odd-number cyclo[n]carbons starting from C 13 .
The molecular structures of the singlet and triplet states of C 11 −C 29 optimized at the CASSCF level have a sharp θ angle at the carbene atom. θ is in the range of 110−123°for the singlet states and 105−140°for the triplet states. The average CC bond lengths at the carbene atom are considerably longer for the CASSCF-optimized molecular structures as compared to the DFT ones ( Figure 2 compared to the singlet state because the p in and p out orbitals are singly occupied. In the triplet state, the electron repulsion is weaker between the p in orbitals but stronger between the p out orbitals than for the singlet state. The singlet states have the p in and p out orbitals formally doubly occupied and nonoccupied, respectively. III.IV. Aromaticity of Odd-Number Cyclocarbons. The structural features of the odd-number cyclo[n]carbons and particularly the θ angle as well as the d 1 and d 2 bond lengths of the carbene moiety determine their aromatic character. Since the lone-pair electron is delocalized at the BHandHLYP/def2-TZVP level, the singlet state of the studied odd-number cyclo[n]carbons is aromatic and sustains a net diatropic ring current according to the double aromaticity nature proposed by Fowler et al. 27 The triplet state, which is the ground state for C 13 13 , C 17 etc.), while the participation of the π in electrons in the global delocalization is prevented by the small θ angle. This is clearly seen in the magnetically induced ring currents shown in Figure 2c and Table S4. The formally empty p out orbital of the [4k+3] cyclo[n]carbons is partially occupied due to interactions with the neighboring p out orbitals. In this case, the carbene atom is less isolated because θ is larger and d 1 /d 2 are shorter than for the [4k+1] rings. Thus, the [4k+2] π out electrons contribute to the global diatropic out-of-plane current and a weaker paratropic ring current inside the ring (Figure 2c). The [4k+3] rings (C 11 , C 15 , C 19 , etc.) are therefore globally aromatic with a low-lying singlet ground state. Since the [4k+3] rings are the most stable molecules among the studied ones, they could be the main target for future synthesis attempts.
III.V. CASSCF and CASPT2 Calculations for C 11 and C 13 . Similar to the case of even-number cyclocarbons, 9 the CASSCF/6-31G(d,p) optimization for all the studied cyclocarbons predicts a significant contribution by only one main determinant into the wave function of the singlet and triplet states. The weights of the [2222222.....] and [22222211....] determinants are close to 0.9 for singlet and triplet states, respectively. Here "2" means doubly occupied orbital, "1" is a singly occupied orbital and "." (dot) is an unoccupied orbital. Structure optimization of the C 11 molecule in singlet and triplet states at CASPT2/SVP level provides similar orbital occupations as the CASSCF/6-31G(d,p) calculation. with weights of about 0.1. Thus, the nondynamic (static) correlation effects for odd-number cyclocarbons are small, but could nevertheless be important. In our previous study on even-numbered cyclocarbons, we showed that static correlation effects decrease with increasing ring size. For C 11 , our CASPT2 calculations predict a negative ΔE ST of −4.7 kcal mol −1 as also obtained at the CASSCF level.
The Journal of Physical Chemistry A pubs.acs.org/JPCA Article However, the CASPT2 value is significantly smaller than the one of −40.2 kcal mol −1 obtained at the CASSCF/6-31G(d,p) level. The magnetically induced ring currents of I(S) = 30 nA T −1 and I(T) = −1.0 nA T −1 calculated at BHandHLYP/ def2-TZVP level using CASPT2/SVP-optimized geometries qualitatively agree with the ones of I(S) = 12 nA T −1 and I(T) = 0.9 nA T −1 obtained when using CASSCF/6-31G(d,p)optimized geometries. The stronger ring current for the singlet state of the CASPT2/SVP-optimized C 11 structure originates from the much smaller θ angle of only 70°between the d 1 and d 2 bonds at the carbene atom as compared to 110°for CASSCF/6-31G(d,p) geometry. Thus, the two atoms next to the carbene atom are much closer in the CASPT2/SVP structure (1.589 Å) than in the CASSCF/6-31G(d,p) geometry (2.204 Å). The CASPT2 calculation confirms the idea that avoiding the empty p out orbital on the carbene carbon atom results in a [4n+2] aromatic π out system, leading to a global aromaticity of the C 11 molecule in the singlet ground state.
To exclude eventual errors related to the single-reference picture of the electronic structure provided by DFT, we performed calculations of the magnetic shielding tensors for the singlet and triplet states of the C 11 and C 13 compounds at the CASSCF level. Integrating the zz component of the magnetic shielding tensor along the symmetry axis in the middle of the ring yields the strength of the magnetically induced ring current according to Ampere−Maxwell's law. 53 The ring-current strength calculated at the BHandHLYP and CASSCF levels for the singlet state of C 11 agree, whereas the CASSCF calculation suggests a stronger ring current for the triplet state than that obtained at the BHandHLYP level. Both levels of theory suggest that the singlet and triplet states of C 13 are antiaromatic. The ring-current strengths calculated at the BHandHLYP level are 2−4 times larger that the CASSCF values (Table 1).
III.VI. Reactivity of Triplet-State Odd-Number Cyclocarbons. Despite the fact that the triplet state of the oddnumber cyclo[n]carbons is generally nonaromatic with an odd number of electrons in the π in and π out subsystems, it has a bond-shift rearrangement between the resonant one (I)-and two (II)-center carbene structures, as seen in Figure 3. A similar bond-shift transformation was previously predicted for the even-number cyclo[n]carbons 10,15 through the cumulenic transition state (TS). For the odd-number cyclo[n]carbons C 15 and C 19 , the TS structure is more complicated, involving a simultaneous change of the θ angle and single−triple bond shifts.
We have successfully optimized both type-I and type-II structures for C 15 and C 19 cyclocarbons in the triplet state. Type-I structures are characterized by a considerably small θ angle (near 120°, Table 2) and a spin density predominantly localized on the carbene atom (Figure 3), while type-II structures sustain a θ angle close to 180° (Table 2) and the spin density equally localized on two atoms adjacent to formal carbene centrum (Figure 3).
The type-I and type-II resonance structures should thus have different reaction pathways and products. The dimerization of the one-center C 15 carbene yields a dimer linked via a double bond, whereas dimerization of the two-center structure C 15 results in a dimer where the two rings are coupled by three CC bonds. It is analogous to the dual behavior of bis(9anthryl)carbene that acts as a two-center biradical when the terminal >C • H groups are unprotected, whereas it acts as a one-center carbene when these groups are protected by phenyl substituents (>C • Ph). 60 The type-I dimers of C 15 and C 19 are predicted to be globally weakly aromatic, sustaining a magnetically induced ring current flowing through the d 1 bond, whereas the type-II dimers of C 15 and C 19 are nonaromatic, sustaining a weak ring current. The rectangular bridge consisting of six carbon atoms sustains local diatropic ("aromatic") ring currents of 6.7 and 5.9 nA T −1 for the dimers of C 15 and C 1 , respectively (Table S5).

IV. CONCLUSIONS
In summary, the electronic and molecular structures of the odd-number cyclo[n]carbons with n = 5, 7, 9, ..., 29 have been studied at DFT (BHandHLYP) and CASSCF levels of theory. The aromatic character and the degree of aromaticity were analyzed by calculating the magnetically induced current density using the GIMIC method. The calculations show that the DFT level of theory predicts a triplet ground state for large odd-number cyclo[n]carbons with n > 11. The molecular structure of the triplet state has polyyne-type CC bonds with